European option pricing with transaction costs and stochastic
volatility: an asymptotic analysis
R. E. Caflisch
Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA rcaflisch@ipam.ucla.edu
G. Gambino and M. Sammartino
Department of Mathematics, University of Palermo, Palermo, Italy gaetana@math.unipa.it marco@math.unipa.it
and C. Sgarra∗
Department of Mathematics, Politecnico di Milano, Milan, Italy ∗Corresponding author: carlo.sgarra@polimi.it
[Received on 3 April 2013; revised on 12 November 2013; accepted on 2 June 2014]
1. Introduction
The intrinsic limitations of the Black–Scholes model in describing real markets behaviour are very well known. Among the main assumptions underlying this model, the most relevant ones are probably constant volatility and no transaction costs. In this paper, we are going to consider the pricing problem of a European option in a model in which both proportional transaction costs are taken into account and the volatility is assumed to evolve according to a stochastic process of the Ornstein–Uhlenbeck type. To analyse this situation we shall follow a utility maximization procedure, following the seminal paper of Davis et al. (1993).
If one uses the following utility functionU:
U(x) = 1 − exp (−γ x),
where γ expresses the risk aversion of the investor, one gets, as the result of this analysis, a non-linear partial differential equation (PDE) for the expected value of the utility-maximized wealth held in the underlying asset of the option.
The pricing of a European option in the presence of small transaction costs was considered in Whalley & Wilmott (1997), where a correction term to the Black and Scholes pricing formula was
derived. This correction term was found to be O(ε2/3), where ε denotes the asymptotic expansion param-eter. Moreover in Whalley & Wilmott (1997) it was found that the optimal hedging strategy consists in not transacting when the process driving the stock price is in a strip around the classical Black and Scholes delta—hedging formula, and in rebalancing the portfolio (selling or buying stocks) to keep the process inside the strip of no transaction. The width of the no transaction strip was found to be O(ε2/3). In Whalley & Wilmott (1997), the volatility has been supposed to be constant.
A wide literature is available on the optimal consumption-investment problem for models with trans-action costs: we just mention two pioneer contributions on the subject, the paper by Davis & Norman (1990), and the paper by Cvitanic & Karatzas (1996). More recently, ksendal & Sulem (2002) inves-tigated the optimal consumption problem in a model including both fixed and proportional transac-tion costs, while Muthuraman & Kumar (2006) studied the optimal investment problem in a multi-dimensional setting. The problem of hedging contingent claims for models with transaction costs has been investigated in several papers: Albanese & Tompaidis (2008), Clewlow & Hodges (1997), Kallsen & Muhle-Karbe (2013) and Zakamouline (2006a,b).
The pricing of a European option with fast mean-reverting stochastic volatility was considered in a series of papers (see e.g. Fouque et al., 2000, 2003, 2001, 2011; Fouque & Lorig, 2011). In the above-mentioned papers, the authors found the pricing formula whose leading order term is the classical Black and Scholes formula with averaged volatility. The correction term was order the square root of the characteristic time scale of the process driving the volatility. An optimal consumption-investment problem has been investigated in a paper by Bardi et al. (2010) where a rigorous asymptotic analysis is performed and where the solution is characterized in the limit of fast volatility dynamics.
In a recent paper, Mariani et al. (2012) proposed a numerical approximation scheme for European option prices in stochastic volatility models including transaction costs based on a finite-difference method. The stochastic volatility dynamics assumed there is a slight generalization of that proposed by Hull & White (1987), since they consider a drift coefficient which is a general (regular) deterministic function of both the time and the underlying asset price, while their diffusion coefficient is linear in the instantaneous volatility.
In the present paper, we propose a different approximation method for European option pricing in stochastic volatility models with transaction costs, based on an asymptotic analysis which follows the approach pioneered by Fouque et al. (2001). The model we consider for the stochastic volatility dynam-ics is of Ornstein–Uhlenbeck type. This model has been originally proposed by Stein & Stein (1991). We provide closed-formulas for the first terms appearing in the asymptotical expansion for European option prices in the limit of fast volatility and small transaction costs. In the present paper, we are not going to provide existence and uniqueness results for the stochastic optimal control problem arising in our pricing methodology, neither we shall discuss rigorously the questions related to asymptotic expansion convergence to the solution required.
The goal of the present investigation is to show how the approach pioneered by Davis et al. (1993) can be extended to stochastic volatility models. To this end we combine with a suitable scaling the asymptotic technique introduced by Whalley & Wilmott (1997) with that proposed by Fouque et al. (2001) in order to obtain quite explicit results. The existence and uniqueness of the solution for the singular control problem arising in American option pricing and in the same modelling framework considered in the present paper has been proved by Cosso & Sgarra (2013). We point out that a recent paper by Bichuch (2011) deals with contingent claim pricing in models with (proportional) transaction costs via an asymptotic analysis based on the pioneer work by Whalley & Wilmott (1997) by providing a rigorous derivation of the asymptotic expansion together with lower and upper bounds for the value function involved. In a previous paper, the same author presented a rigorous asymptotic analysis for
an optimal investment problem, in the case of a power utility function, in finite time for a model with proportional transaction costs (Bichuch, 2012). An approximate hedging strategy construction has been also proposed quite recently in a paper by Lepinette & Quoc (2012) for a local volatility model with proportional transaction costs.
The plan of the paper is the following: in Section 2 we introduce the model considered. In Section 3 we formulate the European option pricing problem as a stochastic control problem, and present the associated HJB equation. In Section 4, the asymptotic analysis is performed, by assuming small trans-action costs and fast mean-reverting volatility. In Section 5, the price of the option is computed and in Section 6, the numerical results and the conclusions are provided. For the reader’s convenience, in Appendix A the source term of the equation obtained through the asymptotic analysis at O(ε) is cal-culated; in Appendix B the averages with respect to the Ornstein–Uhlenbeck invariant measure using the stochastic volatility proposed by Chesney & Scott (1989) are presented; finally, in Appendix C the derivatives, which appears in the obtained corrected pricing formula, with respect to the stock price of the classical Black and Scholes solution are recalled and collected.
2. A stochastic volatility model with transaction costs
We suppose to have the following multi-dimensional stochastic process:
dB= rB dt − (1 + λ)S dL + (1 − μ)S dM , (2.1)
dy= dL − dM , (2.2)
dS= S(α dt + f (z) dW), (2.3)
dz= ξ(m − z) dt + β(ρ dW +1− ρ2dZ). (2.4) In the above equations B and S are the risk-free (the ‘Bond’) and the risky asset (the ‘Stock’), respectively, r is the risk-free interest rate, α is the drift rate of the stock, λ and μ are the (propor-tional) cost of buying and selling a stock, f is the volatility function, that we shall suppose to depend on the stochastic variable z, which is sometimes called the volatility driving process. L(t) and M (t) are the cumulative number of shares bought or sold, respectively, up to time t. Both L(t) and M (t) are assumed to be right continuous with left-hand limits, non-negative and non-decreasing Ft-adapted pro-cesses, where Ftdenotes the filtration generated by (Wt, Zt). Moreover, by convention, we shall assume L(0) = M (0) = 0. We keep the notations introduced in Davis et al. (1993) and Whalley & Wilmott (1997), where the reader can find a detailed justification for the transaction costs model just intro-duced. The set T of trading strategies π(t), in the present setting, consists of all the two-dimensional, right-continuous, measurable processes (Bπ(t), yπ(t)) which are the solution of (2.2) and for which some pair of right-continuous, measurable, Ft-adapted, increasing processes L(t), M (t) exist, such
that(Bπ(t), yπ(t), S(t)) ∈ EΓ ∀t ∈ [0, T], where EΓ= (B, y, S) ∈ R × R × R+: B+ c(y, S) > −Γ , with Γ positive constant and c(y, S) the liquidated cash value of the position y, i.e. the residual cash value when a long position(y > 0) is sold or a short position (y < 0) is closed. Conventionally, we assume c(0, S) = 0. In what follows we shall always suppose f (z) to be a function bounded away from 0:
0< m1 f (z) m2< ∞, ∀z.
The process followed by the stochastic variable z is a Ornstein–Uhlenbeck process with average m. The parameterξ is the rate of mean-reversion volatility.
The Brownian motions W and Z are uncorrelated and ρ is the instantaneous correlation coefficient between the asset price and the volatility shocks. Usually one considers ρ < 0, i.e. the two processes are anti-correlated (e.g. when the prices go down the investors tend to be nervous and the volatility raises). For more details see Fouque et al. (2001) and Jonsson & Sircar (2002).
If we suppose to deal with trading strategies absolutely continuous with respect to time, we can write our stochastic dynamics in the following compact form:
L= t 0 l ds, M= t 0 m ds.
Therefore the process we are dealing with can be written in the following form:
dB= [rB − (1 + λ)Sl + (1 − μ)Sm] dt, (2.5)
dy= (l − m) dt, (2.6)
dS= S(α dt + f (z) dW), (2.7)
dz= ξ(m − z) dt + β(ρ dW +1− ρ2dZ). (2.8) Remark 2.1 The assumption of absolute continuity with respect to time on the processes L(t), M (t) was made in order to write our dynamics in a simpler form, but it can be relaxed. Following the treatment presented in Davis et al. (1993), all the results we are going to mention hold true for L(t) and M (t) right continuous with left-hand limits, non-negative, non-decreasing and Ft-adapted.
If we denote the value at time t of a portfolio of the writer of a European option with strike price K, after following the strategy π, by Φw(t, Bπ(t), yπ(t), S(t), z(t)), its final value is given by:
Φw(T, Bπ(T), yπ(T), S(T), z(T)) = Bπ(T) + I
(S(T)<K)c(yπ(T), S(T))
+ I(S(T)>K)[c(yπ(T) − 1, S(T)) + K]. (2.9) On the other hand, the final value of a portfolio which does not include the option (denoted byΦ1) is simply
Φ1(T, Bπ(T), yπ(T), S(T), z(T)) = Bπ(T) + c(yπ(T), S(T)). (2.10)
3. The optimal control problem for European option pricing
The purpose of this section is to briefly recall the basic ideas of Utility Indifference Pricing and to resume, in a synthetic way, the framework proposed by Davis et al. (1993) for European option pricing in market models with transaction costs. We define the following value functions:
Vj(B) = sup
π∈TE(U(Φj(T, B
π(T), yπ(T), S(T), z(T)))) (3.1)
for j= 1, w. U : R → R is the utility function, which is required to be concave and increasing and satis-fyingU(0) = 0. Note how these value functions depend on the initial endowment B.
Following Davis et al. (1993) we now define:
Bj= inf{B : Vj(B) 0}.
In the present setting the price of the option C, i.e. the amount of money that the writer has to receive to accept the obligation implicit in writing the option, will therefore be:
C= Bw− B1. (3.2)
For this price the investor would in fact be indifferent between the two possibilities of going into the market to hedge the option, or of going into the market without the option. We point out that the present approach, named Utility Indifference Pricing, provides just a criterion to select one price, among the infinitely many compatible with the no-arbitrage requirement, since the present market model is incomplete.
We can define the following function that will be useful in the sequel
Ψj(T, Bπ(T), yπ(T), S(T), z(T)) = Φj(T, Bπ(T), yπ(T), S(T), z(T)) − Bπ(T). (3.3) We have to find an equation for Vj. In what follows we shall suppress the index j and denote Vjwith
V . The problem we are dealing with is a stochastic control problem, where the control is the trading strategy m and l.
By following step by step the procedure illustrated in Davis et al. (1993, pp. 476–477), including the dependence of just one more state variable, it is possible to show that V must satisfy an HJB equation. A rigorous derivation of the (HJB) equation and the existence and uniqueness proof of a solution in the same modelling context, but for the more complex case of American option pricing, can be found in Cosso & Sgarra (2013).
Proposition 3.1 Let the stochastic process be described by (2.5–2.8), then the value function V defined before must satisfy the following (HJB) equation:
max{(∂yVj− (1 + λ)S∂BVj), −(∂yVj− (1 − μ)S∂BVj), ∂tVj+ rB∂BVj+ αS∂SVj+ ξ(m − z)∂zVj +1 2[f(z)] 2S2∂SSVj+1 2β 2∂zzVj+ βfSρ∂SzVj} = 0. (3.4) We now consider the case of the exponential utility functionU(x) = 1 − exp (−γ x). We note, just in passing by, that this gives for Vjthe following expression:
Vj= 1 − inf{E[exp (−γ B(T)) exp (−γ Ψj)]}, whereΨjhas been previously introduced.
In the above maximization problem let us change the variables passing Vj−→ Qj: Vj= 1 − exp
−γδ(B + Qj), where
Note that with the above expression for Vj, the price of the option C, as given in (3.2), now
becomes C = Q1 − Qw(3.5) and we can state the auxiliary result:
Proposition 3.2 The maximization problem for Vjis equivalent to the following minimization problem for Qj: min (∂yQj− (1 + λ)S), (−∂yQj+ (1 − μ)S), ∂tQj− rQj+ αS∂SQj+ ξ(m − z)∂zQj + 1 2[f(z)] 2S2∂SSQj−γ δ(∂SQj)2 +1 2β 2∂zzQj−γ δ(∂zQj)2 + βfSρ∂SzQj−γ δ∂SQj∂zQj = 0. (3.6)
As shown in the papers by Davis et al. (1993) and Whalley & Wilmott (1997), the European option pricing problem is a free boundary problem and the (S, y, z) space divides into three regions, the Buy region, the Sell region and the No Transaction (NT from now on) region. The NT region is separated from the other two regions by two (unknown) boundaries and the optimal policy of the option writer consists in maintain his portfolio in the NT region; when an eventual movement of the asset price forces the portfolio to hit one of the two boundaries, he must trade so as to stay inside the NT region. In the Buy region the following condition must hold
min(∂yQj− (1 + λ)S) = 0, (3.7)
while in the Sell region the following holds:
min(∂yQj− (1 − μ)S) = 0. (3.8)
The NT region is characterized by the following equation: ∂tQj− rQj+ αS∂SQj+1 ε(m − z)∂zQj+ 1 2[f(z)] 2S2∂SSQj−γ δ(∂SQj)2 +1 εν2 ∂zzQj−γ δ(∂zQj)2 +√1 εν √ 2fSρ∂SzQj−γ δ∂SQj∂zQj = 0. (3.9)
Some continuity conditions for the unknown function Q and its first and second derivatives across the free boundaries must be imposed in order to solve the free boundary problem. We shall present them in next section.
4. Small transaction costs and fast mean-reverting volatility: the asymptotic analysis
We now suppose small transaction costs and fast mean-reverting volatility. Moreover, we will assume that the transaction costs are much smaller than the rate of mean reversion:
λ = ¯λε2, μ = ¯με2, ξ =1 ε, β = √ 2ν √ ε .
Buying and selling costs are assumed to be the same for simplicity in the following, whenever anything different will be specified.
We believe that our asymptotic assumptions are consistent with a situation where a large investor, facing very small transaction costs, is involved. In fact, in the empirical study (Fouque et al., 2000), it is found that ε ∼ .005. In the literature (Whalley & Wilmott, 1997; Davis et al., 1993; Zakamouline, 2006a), typically it is assumed 0.2% λ .01%.
In the absence of transaction costs and with a deterministic volatility ε = 0, the investor would con-tinuously trade and get a perfect hedge staying at y = y∗, the ‘B&S’ hedging strategy. When transaction costs are present there is a strip of small thickness around y = y∗ where he does not transact. To resolve this strip we introduce the inner scaled coordinate Y
y= y∗+ εaY and ∂y−→ ε−a∂Y. (4.1)
The unknown boundaries between the NT region and the buy and sell regions are located at: y= y∗+ εaY+ and y= y∗− εaY−.
It is very important from the practical hedger point of view to determine Y+ and Y−.
We impose the following matching conditions (see e.g. Whalley & Wilmott, 1997)
QNT(Y = Y±) = Q(y = y∗± εaY±) continuity,
∂YQNT(Y = Y±) = εa∂yQ(y = y∗± εaY±) continuity of the first derivative, ∂YYQNT(Y = Y±) = ε2a∂yyQ(y = y∗± εaY±) smooth pasting boundary condition.
These boundary conditions will force, in the asymptotic analysis below, a=13. Therefore the NT strip will have a thickness O(ε1/3).
(4.2)
(4.3) In the buy region (Y < Y−) we have (3.7) and a solution is
Q = (1 + λ)Sy + H−(t, S, λ).
In the sell region (Y > Y+) we have (3.8), which solves to
Q = (1 − μ)Sy + H+(t, S, λ).
Here H+(t, S, λ) and H−(t, S, λ) are arbitrary functions not depending on y.
In the NT region we have (3.9), whose solution will be obtained and illustrated in the following
sections.
4.1 The solution in the NT region
As we mentioned before, in the NT region we use the rescaled variable Y defined by (4.1). The change
of variable leads to the following transformation rules for the derivatives: ∂y−→ ε−1/3∂Y, ∂S−→ ∂S− ε−1/3y∗S∂Y, ∂t−→ ∂t− ε−1/3y∗ t∂Y, ∂z−→ ∂z− ε−1/3y∗ z∂Y.
By writing the solution in the NT region in the following form: QNT= S(y∗+ ε1/3Y) + U0(S, t, z) + 13 i=1 εi/6Ui(S, t, z) + ε14/6U 14(S, t, z, Y) + · · · , (4.4)
we can easily obtain an expression for all the derivatives of QNTwith respect to the variables t, S, z. By adopting an obvious notation they can be written as follows:
∂tQNT= U0t+ 11 i=1 εi/6 Uit+ ε12/6(U12t− y∗tU14Y) + · · · , ∂SQNT= y∗+ U0S+ ε1/6U1S+ ε2/6(Y + U2S) + 11 i=3 εi/6 UiS+ ε12/6 U12S− y∗SU14Y + · · · , ∂SSQNT= U0SS+ 9 i=1 εi/6 UiSS+ ε10/6(U10S+ (y∗S)2U14YY) + · · · , ∂zQNT= U0z+ 11 i=1 εi/6 Uiz+ ε12/6(U12z− y∗zU14Y) + · · · , ∂zzQNT= U0zz+ 9 i=1 εi/6 Uizz+ ε10/6(U10zz+ (y∗z)2U14YY) + · · · , ∂SzQNT= U0Sz+ 9 i=2 εi/6 UiSz+ ε10/6(U10Sz+ y∗Sy∗zU14YY) + · · · (4.5)
The introduction of the new scaled variables allows to split the description of the Black–Scholes delta-hedging strategy from the effects due to transaction costs and stochastic volatility and to consider separately their contribution.
To calculate the price of the option we shall use (3.5). The price will have the same asymptotic
expansion as Qjwith j = 1, w, namely C= C0+ ε1/6C1+ ε1/3C2+
√
εC3+ ε2/3C4+ ε5/6C5+ εC6+ · · · . (4.6) Each Ciis given by:
Ci= Ui1− Uiw.
To find the appropriate final conditions for the Ci, we write the final conditions for Q1and Qw. They are, respectively:
Q1(T) = y(T)S(T) (4.7)
and
Given the expression (4.4) one has that the final conditions for the Uiare the following:
Ui1(T) = 0 for i = 0, . . . , 6. (4.9)
U0w(T) = − max(S(T) − K, 0), (4.10)
Uiw(T) = 0 for i = 1, . . . , 6.
Remark 4.1 We point out that the final conditions of our original problem have been imposed on the leading order term, while the corresponding conditions imposed for lower-order terms are homoge-neous. This choice seems to be quite natural in the present setting and is also common for asymptotic expansions.
4.2 The O(ε−1) up to O(ε−1/6) order equations
To simplify the notation, and following the use in Fouque et al. (2003) and Jonsson & Sircar (2002), we define the linear operators Liand the non-linear operator NL:
L0U= (m − z)Uz+ ν2Uzz, (4.11) L1U= −ν √ 2ρ(α − r) f Uz+ ν √ 2fSρUSz, (4.12) L2U= Ut+ 1 2f 2S2USS− rU + rSUS, (4.13) NLU= −ν2γ δ(Uz)2. (4.14)
The O(ε−1) equation is simply
L0U0+ NLU0= 0. (4.15)
The above equation can be considered an ordinary differential equation (ODE) in z for U0: ν2U
0zz+ (m − z)U0z− ν2 γ
δ(U0z)2= 0. (4.16)
In Jonsson & Sircar (2002), it is proved that the only solution of an equation of this form is a U which does not depend on z. The conclusion we therefore draw is that
U0does not depend on z. Analogously, the O(ε−i/6) equations, for i = 1, . . . , 5 are
L0Ui= 0. The conclusion is
4.3 The O(1) equation The O(1) equation writes
L0U6+ L2U0+ S(α − r)(U0S+ y∗S) − 1 2[f(z)]
2S2γ
δ(y∗+ U0S)2= 0. (4.17) The above equation will be analysed in Section 4.5.
4.4 The O(ε1/6) equation The O(ε1/6) equation is
L0U7+ L2U1+ S(α − r)U1S− [f (z)]2S2 γ
δU1S(y∗+ U0S) = 0. (4.18) Also (4.18) will be analysed in Section 4.5.
4.5 The O(ε2/6) equation The O(ε2/6) equation is L0U8+ U2t− r(SY + U2) + αS(Y + U2S) + 1 2[f(z)] 2S2U 2SS− 2 γ δ(y∗+ U0S)(Y + U2S) = 0. (4.19) In the above equation there are terms that do not depend on Y , and terms linear in Y . They must be equal to zero separately. From the terms linear in Y one gets:
y∗= −U0S+
(α − r)δ
f2Sγ . (4.20)
The above expression gives the leading order (in the absence of transaction costs) optimal hedging strategy. One recognizes the Black and Scholes Delta-hedging strategy. The same result appears in the paper by Whalley & Wilmott (1997, formula (3.6)).
By inserting the above expression into the O(1) (4.17), one gets:
L0U6+ ∂tU0+ 1 2f 2S2∂SSU 0+ rS∂SU0− rU0+ 1 2 δ γ (α − r)2 f2 = 0. (4.21) (4.22)
The above equation, considered as an ODE for U6, is of the form:
L0U = χ.
In Fouque et al. (2003) it is shown that the solvability condition for (4.22) is
where the average · is taken with respect to the Ornstein–Uhlenbeck process invariant measure (see Klebaner, 1998 and Fouque et al., 2003, formula (3.3), p. 1652)
χ = 1 ν√2π Rχ(z) e −(m−z)2/2ν2 dz. (4.24)
Therefore the solvability condition for (4.21) is
∂tU0+ 1 2¯σ 2S2∂SSU 0− rU0+ rS∂SU0= − δ(α − r)2 2γ 1 ¯τ2, (4.25)
where ¯σ is the effective constant volatility
¯σ2= f2 , and¯τ is defined as 1 ¯τ2= 1 f2 .
Once one imposes to (4.25) the appropriate final condition, which will be different for the investor with option liability and the investor without it, then U0 is determined. One can go back to equation
(4.21) and solve it for U6. Once (4.21) has been solved, and the proper boundary conditions are taken
into account, it is easy to recognize that the solution for U6 can be written as a superposition of two
terms, one independent on z (the differential operator L2 contains only terms proportional to the first
derivatives with respect to z and this term satisfies the homogeneous equation L2U = 0) and one
depend-ing on (S, z, t), in such a way that we can write
U6= U6(z)(S, z, t) + ˜U6(S, t), (4.26) where, U6(z)(S, z, t), the part of U6which depends on z, has the following expression
U6(z)(S, z, t) = −L−10 1 2S 2(f2− ¯σ2)U 0SS+ 1 2 δ γ(α − r)2 1 f2 − 1 ¯τ2 ;
on the other hand, ˜U6is a function that does not depend on z and that will be determined by the O(ε) equation in the asymptotic procedure.
We can get a more explicit representation for U6(z), that will be useful in the next subsection. We first define the functionsϕ(z) and ψ(z) as the solutions of the following problems:
L0ϕ = f2− f2 , (4.27) L0ψ = 1 f2 − 1 f2 . (4.28)
Therefore the above expression for U6(z)can be written as:
U6(z)= − 1 2S 2U 0SSϕ + 1 2 δ γ(α − r)2ψ . (4.29)
(4.30)
Let us come back to (4.18). Once substituted the expression (4.20) for y∗, (4.18) reduces to
L0U7 + L2U1 = 0.
Equation (4.30) is a Poisson problem for U7, whose solvability condition reads:
L2 U1= 0, (4.31)
which is a homogeneous Black–Scholes equation for U1. Given that the final condition is zero, we get the conclusions
U1≡ 0,
U7= ˜U7(S, t) does not dependent on z.
One can now go back to (4.19), collect all terms independent of Y and get the following equation: L0U8 + L2U2 = 0.
The above equation is a Poisson problem of the type (4.22). The solvability condition is:
L2 U2= 0. (4.32)
Note that the above equation is homogeneous in U2. Given that the final condition, both for the investor with option liability and for the investor without it, is 0, one gets the following conclusions
U2≡ 0,
U8= ˜U8(S, t) is independent of z.
Therefore, the first significant correction to the Black and Scholes value is the O(ε1/2) contribution.
4.6 The O(ε3/6) equation
Using the expression (4.20) for y∗, the O(ε3/6) equation can be written as
L2U3+ L1U6+ L0U9= 0. (4.33)
˜U
˜
Note that in the above equation appears U6, which until now we have derived only up to the function 6(S, t), to be determined by a higher order asymptotic. However, in (4.33) U6 is hit by the operator L1,
which cancels U6(S, t).
Therefore, one can consider (4.33) as a Poisson problem for U9, whose solvability condition is
L2 U3= − L1U6 . (4.34)
The above equation is a Black and Scholes equation for U3with 0 final condition and with a source term. We now want to rewrite the source term.
By using for U6 the expression (4.26) the operator L1 cancels the part not depending on z, and taking
into account the expression (4.29), one can express the source term in (4.34) as
− L1U6 = L1 1 2S 2U 0SSϕ + 1 2 δ γ(α − r)2ψ =√νρ 2 f ϕ (S3U 0SSS+ 2S2U0SS) − (α − r)S2U0SS ϕ f −γδ(α − r)3 ψ f . Therefore, U3solves the following Black and Scholes equation
U3t+ 1 2¯σ 2S2U 3SS− rU3+ rSU3S =√νρ 2 f ϕ (S3U 0SSS+ 2S2U0SS) − (α − r)S2U0SS ϕ f −γδ(α − r)3 ψ f (4.35) with zero final data.
4.7 The O(ε2/3) equation
Since U7does not depend on z, the O(ε2/3) equation can be written as L2U4+ L0U10+ ν2(y∗z)
2
U14YY − γ
δf2S2Y2= 0. (4.36)
Equation (4.36) can be considered as an ODE in Y for U14. It writes as
U14YY= AY2+ B, (4.37)
where we have defined the following quantities A=γ δ f2S2 ν2(y∗ z)2 , B= −L2U4+ L0U10 ν2(y∗ z)2 . Equation (4.37) solves to U14= A 12Y 4+1 2BY 2+ CY + D, (4.38)
with C and D independent of Y . Now we have to impose the matching conditions. Being
QBUY= (1 + ε2¯λ)Sy + H−(S, z, t) in the outer buy region QSELL= (1 − ε2¯μ)Sy + H+(S, z, t) in the outer sell region
(4.39) and imposing the continuity of the gradient at the two boundaries:
∂YQNT(Y = −Y−) = ε1/3∂yQBUY(y = y∗− ε1/3Y−), ∂YQNT(Y = Y+) = ε1/3∂yQSELL(y = y∗+ ε1/3Y+),
one gets, at O(ε14/6)
∂YU14(Y = −Y−) = ¯λS, (4.40)
∂YU14(Y = Y+) = − ¯μS. (4.41)
Therefore, using (4.38), one gets
−A 3(Y −)3− BY−+ C = ¯λS, (4.42) A 3(Y +)3+ BY++ C = − ¯μS. (4.43)
Moreover, being W in the outer regions linear in y, one imposes the continuity of the second deriva-tive as follows
∂YYQNT(Y = ±Y±) = 0, i.e.
A(Y+)2+ B = 0, A(Y−)2+ B = 0.
From these equations one sees that, at this order, the bandwidth about the Black and Scholes strategy is symmetric, i.e. Y+= Y−= −B A 1/2 . (4.44)
Subtracting the two equations (4.42) and (4.43) to eliminate C, and using the above expressions for Y±,
one gets
4 3(−B)
3/2A−1/2= (¯λ + ¯μ)S.
After some manipulations, and using the expressions for A and B, (4.37) leads to the following equation:
L0U10+ L2U4= 3 4(¯λ + ¯μ)fS 2 γ δν2(y∗z)2 2/3 . (4.45)
One can also find an expression for the amplitude of the NT region
Y+= Y−= 3 2 1 f2S δ γν2(y∗z)2 1/3 . (4.46)
Equation (4.45) is a Poisson problem of the type of (4.22). The solvability condition gives an
equation for U4: L2 U4= 3 4(¯λ + ¯μ)fS 2 γ δν2(y∗z)2 2/3 . (4.47)
Note also that, adding the two equations (4.42) and (4.43), one gets that C = 0. Therefore U14= A 12Y 4+1 2BY 2+ D, (4.48)
which will be useful in Section 4.9. 4.8 The O(ε5/6) equation
The O(ε5/6) equation writes as:
L0U11+ L2U5− √ 2∂U6 ∂z γ δνρf (z)SY − ∂U3 ∂S γ δf(z)2S2Y + ∂2U 15 ∂Y2 ∂y∗ ∂z 2 ν2= 0. (4.49)
This equation can be considered as an ODE for U15: ∂2U
15
∂Y2 = ¯AY + ¯B, where we have denoted
¯A =√2∂U6 ∂z γ δνρf (z)S − ∂U3 ∂S γ δf(z)2S2 (ν2y∗2 z ), ¯B = −L0U11+ L2U5 ν2y∗2 z .
Integrating (4.49) with respect to Y and using the boundary conditions:
U15Y(Y+) = U15Y(−Y−) = 0 which are needed to ensure the continuity of the gradient, one gets
L0U11+ L2U5= 0. (4.50)
The above equation is a Poisson problem for U11, whose solvability condition reads
L2 U5= 0. (4.51)
This is a homogeneous Black–Scholes equation for U5. Given that the final condition is zero, we get the conclusions:
U5≡ 0,
4.9 The O(ε) equation
Collecting the O(ε) terms one gets L2U6+ L1U9+ L0U12− 1 2f 2S2γ δ(U3S)2− ν √ 2fSργ δU3SU6z− y∗z(m − z)U14Y −γδf2YS2U4S+ ν2 −y∗
zzU14Y − 2y∗zU14Yz+ (y∗z)2U16YY− γ δ(U6z)2
= 0. (4.52)
The above equation can be considered as an ODE in Y for U16: ∂2U
16
∂Y2 = ˜AY + ˜B, where we have defined
˜A = γδ f2S2U4S ν2(y∗ z)2 , ˜B = −L0U12+ L1U9+ L2U6− 1 2f 2S2γ δU3S2 − ν √ 2fSργ δU3SU6z − ν2y∗ zzU14Y+ 2y∗zU14Yz+ γ δU6z2 + U14Yy∗z(m − z) (ν2(y∗ z) 2).
Note that ˜A does not depend on Y and in ˜B the Y -dependent terms appear only with their derivatives in Y .
We integrate (4.52) from −Y− to Y+. Let us use the boundary conditions:
U16Y(Y+) = U16Y(−Y−) = 0.
Moreover, being Y+ = Y− and from the expression (4.48) it follows that Y+ −Y−U14YdY= 0. By integrating (4.52) we get: L0U12+ L1U9+ L2U6= 1 2f 2S2γ δ(U3S)2+ ν √ 2fSργ δU3SU6z+ ν2γ δ(U6z)2. The solvability condition for U12gives the following equation for U6:
L2 U6= − L1U9 + 1 2¯σ 2S2γ δ(U3S)2+ ν √ 2Sργ δU3S fU6z(z) + ν2 γ δ (U6z(z))2 . (4.53) The main results of this section are the following:
1. Equation (4.25) for U0;
3. Equation (4.32) for U2 which led us to U2 ≡ 0;
4. Equation (4.35) for U3;
5. Equation (4.47) for U4;
6. Equation (4.51) for U5 which led us to U5 ≡ 0;
7. Equation (4.53) for U6;
8. The expression (4.20) for y∗, the centre of the NT region.
9. The expression (4.46) for the boundaries of the NT region. 5. The option pricing
To calculate the price of the option we now use (3.5), and the asymptotic expansion (4.6) together with the appropriate final conditions for the Ci, which, as we discussed in the previous section, can be obtained by the final conditions for the Ui, (4.9–4.11). Here we can state our main result, which will be proved in detail in the following subsections:
Proposition 5.1 The first seven terms of the asymptotic expansion (4.6) in powers of the adopted parameter ε for the European Call option price in the Utility Indifference framework introduced by M.A.H. Davis, V.G. Panas and T. Zariphopoulou are provided by formulas (5.3), (5.4), (5.5), (5.8), (5.9), (5.10) and (5.15) written below.
5.1 The leading order price
In order to compute the leading order price we have to calculate U01 and U0wwhere they both satisfy
(4.25). Given the respective final conditions (4.9) and (4.10) one has that
U01= (T − t)δ(α − r) 2 2γ 1 ¯τ2, (5.1) U0w= (T − t)δ(α − r) 2 2γ 1 ¯τ2 − C BS, (5.2)
where CBSis the classical pricing formula for a European call option, i.e. CBS(S, t) = SN(d1) − K e−r(T−t)N(d2), where d1= log(S/K) + (r + (1/2) ¯σ2)(T − t) ¯σ√T− t , d2= d1− ¯σ √ T − t,
and N(z) is the normal cumulative distribution function. From the above expressions for U0jone obtains
5.2 The O(ε1/6) correction The equation for U1
1 and U1 wis (
4.31), a homogeneous Black and Scholes equation. In both cases, the
final condition is homogeneous. Therefore U1
1≡ 0 and U1w≡ 0 and
C1(S, t) = 0. (5.4)
5.3 The O(ε1/3) correction
As for the O(ε1/6) terms, the equation for U1 2 and U
w
2 is the homogeneous Black and Scholes equation (4.32), with homogeneous final condition, therefore both U1
2 and U2ware zero and
C2(S, t) = 0. (5.5)
5.4 The O(ε1/2) correction The equation for U1
3 and U3
wis (4.35), in both cases with homogeneous final condition. Using, respec-tively, the expressions (5.1) and (5.2) in (4.35), one has that
U31= (T − t)√νρ 2 δ γ(α − r)3 ψ f , (5.6) U3w= −(T − t)√νρ 2 −δ γ(α − r)3 ψ f − f ϕ (S3CBS 3S + 2S2CSSBS) + (α − r)S2 ϕ f CSSBS . (5.7) Therefore C3(S, t) = −(T − t)√νρ 2 f ϕ (S3∂3 SCBS+ 2S2∂SSCBS) − (α − r)S2∂SSCBS ϕ f . (5.8)
5.5 The O(ε2/3) correction The equation for U1
4 and U w
4 is (4.47), in both cases with homogeneous final condition. The source term for the two problems is the same: in fact y∗z has the same expression for both problems. Therefore U4w= U1
4 and
C4(S, t) = 0. (5.9)
5.6 The O(ε5/6) correction The equation for U1
5 and U5wis the homogeneous Black and Scholes equation (4.51) with zero final condition in both cases. Then
C5(S, t) = 0. (5.10)
5.7 The O(ε) correction
To compute the O(ε) correction we have to solve (4.53). It is a Black and Scholes equation with a source
see (4.26). The same decomposition holds also for C6: C6= C(z)6 + ˜C6, where C6(z)= U6(z)1− U6(z)w, and ˜C6= ˜U6(z)1− ˜U w 6.
We have already computed U6(z)j, as given in (4.29), we can therefore calculate C6(
z). In fact, using (4.29) and the expressions (5.1) and (5.2), one gets
U6(z)1= −1 2 δ γ(α − r)2ψ, (5.11) U6(z)w=1 2S 2CBS SSϕ − 1 2 δ γ(α − r)2ψ, (5.12) which gives C(z)6 = −1 2S 2CBS SSϕ. ˜
We are now left with the task of computing C6.
The equation for C6 can be derived using (4.53). Subtracting the two equations relative to U1 6 and U6wone gets L2 C6= −[ L1U91 − L1U9w ] + 1 2S 2¯σ2γ δ (U1 3S)2− (U w 3S)2 + ν√2Sργ δ U3S1 fU6z(z)1 − U3Sw fU6z(z)w + ν2γ δ U6z(z)1)2 − (U6z(z)w)2 . (5.13) This equation is a Black and Scholes equation for C6 with source term. In Appendix A, this source term is explicitly computed and (5.13) writes as
L2 C6= (T − t)2ˆA + (T − t)ˆB + ˆC, (5.14) where ˆA = −ν2ρ2 4 γ δS2¯σ2 f ϕ (S3CBS 4S + 5S2CBSSSS+ 4SCBSSS) − (α − r) ϕ f (S2CBS SSS+ 2SCBSSS) 2 , ˆB = −ν2ρ2S2 ϕf − 1 2(α − r) ϕ f f ϕ (S3CBS 5S + 8S2C4SBS + 14SCBS SSS+ 4CSSBS) − (α − r) ϕ f (S2CBS 4S + 4SCSSSBS + 2CBSSS)
−ν2ρ2 2 S 3 f ϕ (S3CBS 6S + 11S2C5SBS+ 30SCBS4S + 18CSSS) f ϕBS −(α − r)(S2CBS 5S + 6SC4SBS+ 6CBSSSS) ϕ f −ν2ρ2 2 γ δS f ϕ (S3CBS 4S + 5S2CBS3S + 4SCBSSS)(α − r) ϕ f (2SCBS SS + S2CBS3S) × S2CBSSS ϕf − δ γ(α − r)2 ψf , ˆC = ν2γ δ −1 4S 4(CBS SS) 2 ϕ2 + 1 2 δ γ(α − r)2S2CSSBS ϕψ − ν2ρ2(α − r) (α − r)S2CBS SS G f − (S 3CBS SSS+ 2S 2CBS SS) F f − ν2ρ2(S4CBS 4S + 5S3C4SBS+ 5S3CBSSSS+ 4S2CBSSS) Ff − (α − r)(S3CSSSBS + 2S2CSSBS) Gf . Given the homogeneous final condition, the solution of (5.14) writes as
C6=(T − t) 3 3 ˆA + (
T− t)2
2 ˆB + (T − t) ˆC. (5.15)
We have used the fact that L2 (T − t)3 3 A+ (T − t)2 2 B+ (T − t)C = (T − t)2A+ (T − t)B + C + (T − t)3 3 L2A+ (T − t)2 2 L2B+ (T − t)L2C and the last three terms are zero
L2 Sn∂ nCBS ∂Sn = Sn ∂n ∂SnL2C BS= 0.
In Appendix C, the reader can find the derivatives of CBS with respect to S up to the sixth order. In Appendix B, the averages · with respect to the Ornstein–Uhlenbeck process invariant measure are explicitly computed using Scott’s model.
6. Numerical illustration of the results
In this section, we present the main results obtained via the asymptotic method. At first, we plot the NT region for different values of the volatility, ranging from 5 to 60%, both in the case which does not include the option, denoted by the index 1, and in the case which includes the option, denoted by the index w. The volatility is chosen as in the Scott model, f (z) = ez.InFig. 1, the curves representing the
Black and Scholes strategy y∗ in the absence of transaction costs and the hedging boundaries, y = y∗ ±
0 0.2 0.4 0.6 0.8 1 1.2 0 100 200 300 400 500 600 700 800 900 1000 1100 y 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 30 40 50 60 70 S y S (a) (b) (d) (c) 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 12 14 16 18 S y 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4 5 6 7 8 S y
Fig. 1. The NT region in the case which does not include the option. The dotted curve represents the Black and Scholes strategy
y∗in the absence of transaction costs, the other two curves represent the hedging boundaries. See the text for the choice of parameters. Parts (a,b,c,d) refer to different numerical values of the initial volatility.
that these curves are, respectively, given by
y∗=(α − r)δ e2zSγ , (6.1) y=(α − r)δ e2zSγ ± ε 1/33(¯λ + ¯μ)(α − r)2ν2δ3 e6zS3γ3 1/3 . (6.2)
The corresponding curves in the second case are plotted in Fig. 2and their equations are:
y∗= CBSS +
(α − r)δ
0 1 2 3 4 5 6 0 20 40 60 80 100 120 140 160 180 200 S y 0 1 2 3 4 5 6 0 2 4 6 8 10 12 S y 0 1 2 3 4 5 6 0 0.5 1 1.5 2 2.5 3 (c) (a) (b) (d) S y 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 S y
Fig. 2. The NT region in the case which includes the option. The dotted curve represents the Black and Scholes strategy y∗in the absence of transaction costs, the other two curves represent the hedging boundaries. See the text for the choice of parameters. Parts (a,b,c,d) refer to different numerical values of the initial volatility.
y= CSBS+(α − r)δ e2zSγ ± ε 1/33(¯λ + ¯μ)(α − r)2ν2δ3 e6zS3γ3 1/3 . (6.4)
Both in Figs 1 and 2, the strike price is K = 0.5, the risk-free interest rate is r = 0.07, the drift rate of the stock is α = 0.1, the risk aversion is γ = 1, the mean volatility σ¯ = 0.2, the time to expiry is 0.3, ¯λ = ¯μ = 1 and ε = 1
200.
Finally, in Fig.3it is shown the curve representing the classical Black and Scholes price of a Euro-pean call option with the first correction obtained at O(ε1/2) and the second correction obtained at O(ε). Here the parameters are chosen as K= 100, r = 0.04, α = 0.1, γ = 1, the time to expiry is 0.25 and ε = 1
200. In Fig.3(a,b) the correlation coefficient isρ = 0, therefore the Black and Scholes price and the corrected price at O(ε1/2) coincide as follows from the expression (5.8). The solutions are computed at two levels of the current volatility σ2= 0.165 and σ2= 0.66 and in the range around the money 0.9 S/K 1.1 the maximum deviation of the asymptotic approximation (at O(ε)) from the price with
(a)
(c) (d)
(b)
Fig. 3. Comparison between the classical price for a European call option and the price including the correction up to O(ε1/2)
and up to O(ε), at two levels of the current volatility. (a, b) The region around the money, with ρ = 0 and (c, d) the region out of
the money, with ρ = −0.3.
the lower volatility is 2.1% of this price, with the higher volatility is 7.5%. In Fig. 3(c,d) the correlation coefficient is ρ = −0.3. We note that in the region out of the money the second correction of the price obtained at O(ε) improves the approximation obtained at O(ε1/2). We want to remark that the
oscilla-tory behaviour exhibited in Fig. 2(b–d) by y for small values of S has been already observed by Whalley & Wilmott (1997) also in models with constant volatility, although this feature seems to be more pro-nounced in the present context. Moreover, the thickness of the NT region in the presence of stochastic volatility seems to be bigger than in model with constant volatility. The (ε)1/3 scaling was
also a relevant feature already established in Whalley & Wilmott (1997) which is exhibited also by the present model.
Acknowledgements
Funding
The work by GG was partially supported by GNFM/INdAM through Progetto Giovani Grant.
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Appendix A. The source term in the equation for C6
We separately compute the four terms appearing into the source of (5.13) for C6. A.1 The term L1U91 − L1U9w
In the source term we have [ L1U91 − L1U9w ]. We now write L1U j
9 for j = 1, w, and for notational simplicity we omit the index j.
Consider (4.33) and solve it for U9:
U9= −L−10 (L2U3+ L1U6). (A.1) One can go back to (4.35) for the termL2U3:
L2U3= 1 2S 2U 3SSL0ϕ + νρ √ 2 f ϕ (S3U 0SSS+ 2S2U0SS) − (α − r)S2U 0SS ϕ f −γδ(α − r)3 ψ f .
Being L1U6 = L1U6(z), from (4.26) it follows that
L1U6(z)= νρ √ 2 −f ϕ(2S2U 0SS+ S3U0SSS) + (α − r)S2U0SSϕ f + δ γ(α − r)3 ψ f .
Defining F, G and H as the solutions of the following Poisson equations: L0F(z) = f ϕ− f ϕ , L0G(z) =ϕ f − ϕ f , L0H(z) =ψ f − ψ f ,
the formula (A.1) writes as
U9= − 1 2S 2U 3SSϕ + νρ √ 2 (S3U 0SSS+ 2S2U0SS)F(z) − (α − r)S2U0SSG(z) − δ γ(α − r)3H(z) .
The first term in the right-hand side of (5.13) is L1U91 − L1U9w = ν2ρ2S2(T − t) ϕf −1 2(α − r) ϕ f f ϕ (S3CBS 5S + 8S2CBS4S + 14SCBS SSS+ 4CBSSS) − (α − r) ϕ f (S2CBS 4S + 4SCSSSBS + 2CBSSS) + ν2ρ2(α − r) (α − r)S2CBS SS G f − (S3CBS SSS+ 2S 2CBS SS) F f +ν2ρ2 2 S 3(T − t) f ϕ (S3CBS 6S + 11S 2CBS 5S + 30SC BS 4S + 18C BS SSS) f ϕ − (α − r)(S2CBS 5S + 6SC5SBS+ 6SCBS4S + 6CSSS)BS ϕ f + ν2ρ2(S4CBS 4S + 5S3C4SBS+ 5S3CSSSBS + 4S2CBSSS) Ff − (α − r)(S3CBS SSS+ 2S2CBSSS) Gf . A.2 The term(U1
3S)2− (U w 3S)2
Using the expressions (5.6) and (5.7), respectively, for U1
3 and U3w, it follows that (U1 3S) 2− (Uw 3S) 2= −ν2ρ2 2 (T − t) 2 f ϕ (S3CBS 4S + 5S 2CBS SSS+ 4SC BS SS) − (α − r) ϕ f (S2CBS SSS+ 2SCSSBS) 2 .
A.3 The term U3S1 fU6z(z)1 − U3Sw fU6z(z)w Since U31is independent of S:
U3S1 fU6z(z)1 = 0. Using the expression (4.29), we have
U3Sw fU6z(z)w = νρ 2√2(T − t) f ϕ (S3CBS 4S + 5S2C3SBS+ 4SCBSSS) − (α − r) ϕ f (2SCBS SS + S2CBS3S) × S2CSSBS ϕf − δ γ(α − r)2 ψf .
A.4 The term (U6(z z)1)2
− (U6(z z)w)2
Using once again the formula (4.29) it follows
(U6z(z)1)2− (U6z(z)w)2 = − 1 4S 4(CBS SS)2 ϕ2 + 1 2 δ γ(α − r)2S2CBSSS ϕψ .
Appendix B. Calculation of the averages · for Scott’s model If one uses, for the volatility f(z) the model of Scott, i.e.
f(z) = ez,
then one can compute explicitly
f2 = em+ν2 , 1 f2 = em−ν2 , ϕ f =ν12(em+(1/2)ν2 − em+(5/2)ν2 ), ψ f =ν12(e(3/2)(−2m+3ν2) − em−(3/2)ν2 ), ψf = 1 ν2(e 3(m−(1/2)ν2) − e(1/2)(ν2−2m) ), f ϕ = 1 ν2(e 3m+(5/2)ν2 − e3m+(9/2)ν2 ), f = em+(1/2)ν2 , 1 f = e−m+(1/2)ν2 , f2ϕ = 1 2ν2(e 4(m+ν2) − e4(m+2ν2) ), ϕ f2 = 1 2ν2(1 − e 4ν2 ), ϕ = −2 e2(m+ν2) , fF = 1 ν4 e4m+3ν2− e4m+5ν2+1 2e 4(m+2ν2) −1 2e 4(m+ν2) , fG = 1 ν4(2ν 2e2(m+ν2) + e2m+ν2 − e2m+3ν2 ), F f = 1 ν4(−2ν 2e2(m+ν2) + e2m+5ν2 − e2m+3ν2 ), G f = 1 ν4 1 2− 1 2e 4ν2 + e3ν2 − eν2 , H f = 1 ν4 −1 2 + 1 2e 4(2ν2−m) − e−4m+5ν2 + e−ν2 .
Finally, the term ϕψ can be written as ϕψ = 2 e4ν 2 √πν2 +∞ −∞ e u2
(erf(u −√2ν) − erf(u))(erf(u +√2ν) − erf(u)) du.
The value of this integral is numerically calculated. In particular, forν = 1/√2 we obtain ϕψ = −24.8229.
Analogously, we rewrite ϕ2 in the following form ϕ2 =2 e4(m+ν 2) √πν2 +∞ −∞ e u2 (erf(u −√2ν) − erf(u))2du
and we numerically compute it. Forν = 1/√2, it is ϕ2 = 17.329 e2(2m+1). Appendix C. The derivatives of CBS
The derivatives of CBSwith respect to S, up to the sixth order, are explicitly calculated as follows ∂SCBS= N(d1), ∂SSCBS= e −d2 1/2 S¯σ√2π(T − t), ∂3 SC BS= −e−d 2 1/2 S2¯σ√2π(T − t) 1+ d1 ¯σ√T− t , ∂4 SCBS= e−d12/2 S3¯σ√2π(T − t) d12− 1 ¯σ2(T − t)+ 3d1 ¯σ√T− t+ 2 , ∂5 SCBS= e−d12/2 S4¯σ√2π(T − t) − d1 ¯σ√T− t + 3 d12− 1 ¯σ2(T − t)+ 3d1 ¯σ√T− t+ 2 + 1 ¯σ2(T − t) 2d1 ¯σ√T− t+ 3 , ∂6 SCBS= e−d12/2 S5¯σ√2π(T − t) d2 1 − 1 ¯σ2(T − t)+ 3d1 ¯σ√T− t + 2 d1 ¯σ√T− t+ 3 × d1 ¯σ√T− t + 4 − 1 ¯σ(T − t) + 2 ¯σ4(T − t)2 − ¯σ2(T − t)1 2d1 ¯σ√T− t+ 3 2d1 ¯σ√T− t+ 7 .
